Beyond the triangle and uniqueness relations: non-zeta counterterms at large N from positive knots

TL;DR

This paper explores non-zeta counterterms in large-N quantum field theories, revealing connections to complex knots and hypergeometric series, and provides analytical and numerical results up to high loop orders.

Contribution

It introduces the analysis of non-zeta counterterms linked to positive knots and hypergeometric series, extending understanding of anomalous dimensions at high loop orders.

Findings

Irreducible double sums correspond to specific torus knots in anomalous dimensions.

High-precision numerical results up to 24 loops support analytical calculations.

Knots generated by three dressed propagators relate to order 1/N^3 corrections.

Abstract

Counterterms that are not reducible to $\zeta_{n}$ are generated by ${}_3F_2$ hypergeometric series arising from diagrams for which triangle and uniqueness relations furnish insufficient data. Irreducible double sums, corresponding to the torus knots $(4,3)=8_{19}$ and $(5,3)=10_{124}$, are found in anomalous dimensions at ${\rm O}(1/N^3)$ in the large-$N$ limit, which we compute analytically up to terms of level 11, corresponding to 11 loops for 4-dimensional field theories and 12 loops for 2-dimensional theories. High-precision numerical results are obtained up to 24 loops and used in Pad\'e resummations of $\varepsilon$-expansions, which are compared with analytical results in 3 dimensions. The ${\rm O}(1/N^3)$ results entail knots generated by three dressed propagators in the master two-loop two-point diagram. At higher orders in $1/N$ one encounters the uniquely positive hyperbolic 11-crossing knot, associated with an irreducible triple sum. At 12 crossings, a pair of 3-braid knots is generated, corresponding to a pair of irreducible double sums with alternating signs. The hyperbolic positive knots $10_{139}$ and $10_{152}$ are not generated by such self-energy insertions.